

A Smarandache StrongWeak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure. By proper subset of a set S, we mean a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any. In any field, a Smarandache strongweak nstructure on a set S means a structure {w_{0}}
on S such that there exist two chains of proper subsets P_{n1 }
<
P_{n2 }
< …
<
P_{2 }< P_{1
}<
S and Q_{n1 }
<
Q_{n2 }
< …
<
Q_{2 }<
Q_{1
}<
S, where '<' means 'included in', whose corresponding stronger structures verify the chain {w_{n1}} >
{w_{n2}} >
… >
{w_{2}} >
{w_{1}} >
{w_{0}}
and respectively the weaker structures verify the chain {v_{n1}} < {v_{n2}} <
… < {v_{2}} < {v_{1}} < {v_{0}},
where '>'
signifies 'strictly stronger' (i.e. structure satisfying more axioms) and '<'
signifies 'strictly weaker' (i.e. structure satisfying less axioms). And by structure on S we mean a structure {w} on S under the given operation(s). As a particular case, a Smarandache strongweak 2structure (two levels only of structures in algebra) on a set S, is a structure {w_{0}} on S such that there exist two proper subsets P and Q of S, where P is embedded with a stronger structure than {w_{0}}, while Q is embedded with a weaker structure than {w_{0}}. For example, a Smarandache strongweak monoid is a monoid that has a proper subset which is a group, and another proper set which is a semigroup. Also, a Smarandache strongweak ring is a ring that has a proper subset which is a field, and another proper subset which is a nearring.
